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Theorem fzsplit3 23352
Description: Split a finite interval of integers into two parts. (Contributed by Thierry Arnoux, 2-May-2017.)
Assertion
Ref Expression
fzsplit3  |-  ( K  e.  ( M ... N )  ->  ( M ... N )  =  ( ( M ... ( K  -  1
) )  u.  ( K ... N ) ) )

Proof of Theorem fzsplit3
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elfzelz 10890 . . . . . . 7  |-  ( x  e.  ( M ... N )  ->  x  e.  ZZ )
21zred 10209 . . . . . 6  |-  ( x  e.  ( M ... N )  ->  x  e.  RR )
3 elfzelz 10890 . . . . . . . 8  |-  ( K  e.  ( M ... N )  ->  K  e.  ZZ )
43zred 10209 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  K  e.  RR )
5 1re 8927 . . . . . . . 8  |-  1  e.  RR
65a1i 10 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  1  e.  RR )
74, 6resubcld 9301 . . . . . 6  |-  ( K  e.  ( M ... N )  ->  ( K  -  1 )  e.  RR )
8 lelttric 9017 . . . . . 6  |-  ( ( x  e.  RR  /\  ( K  -  1
)  e.  RR )  ->  ( x  <_ 
( K  -  1 )  \/  ( K  -  1 )  < 
x ) )
92, 7, 8syl2anr 464 . . . . 5  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  -> 
( x  <_  ( K  -  1 )  \/  ( K  - 
1 )  <  x
) )
10 elfzuz 10886 . . . . . . 7  |-  ( x  e.  ( M ... N )  ->  x  e.  ( ZZ>= `  M )
)
11 1z 10145 . . . . . . . . 9  |-  1  e.  ZZ
1211a1i 10 . . . . . . . 8  |-  ( K  e.  ( M ... N )  ->  1  e.  ZZ )
133, 12zsubcld 10214 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  ( K  -  1 )  e.  ZZ )
14 elfz5 10882 . . . . . . 7  |-  ( ( x  e.  ( ZZ>= `  M )  /\  ( K  -  1 )  e.  ZZ )  -> 
( x  e.  ( M ... ( K  -  1 ) )  <-> 
x  <_  ( K  -  1 ) ) )
1510, 13, 14syl2anr 464 . . . . . 6  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  -> 
( x  e.  ( M ... ( K  -  1 ) )  <-> 
x  <_  ( K  -  1 ) ) )
16 elfzuz3 10887 . . . . . . . . 9  |-  ( x  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  x )
)
1716adantl 452 . . . . . . . 8  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  ->  N  e.  ( ZZ>= `  x ) )
18 elfzuzb 10884 . . . . . . . . 9  |-  ( x  e.  ( K ... N )  <->  ( x  e.  ( ZZ>= `  K )  /\  N  e.  ( ZZ>=
`  x ) ) )
1918rbaib 873 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  x
)  ->  ( x  e.  ( K ... N
)  <->  x  e.  ( ZZ>=
`  K ) ) )
2017, 19syl 15 . . . . . . 7  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  -> 
( x  e.  ( K ... N )  <-> 
x  e.  ( ZZ>= `  K ) ) )
21 eluz 10333 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  x  e.  ZZ )  ->  ( x  e.  (
ZZ>= `  K )  <->  K  <_  x ) )
223, 1, 21syl2an 463 . . . . . . 7  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  -> 
( x  e.  (
ZZ>= `  K )  <->  K  <_  x ) )
233adantr 451 . . . . . . . 8  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  ->  K  e.  ZZ )
241adantl 452 . . . . . . . 8  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  ->  x  e.  ZZ )
25 zlem1lt 10161 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  x  e.  ZZ )  ->  ( K  <_  x  <->  ( K  -  1 )  <  x ) )
2623, 24, 25syl2anc 642 . . . . . . 7  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  -> 
( K  <_  x  <->  ( K  -  1 )  <  x ) )
2720, 22, 263bitrd 270 . . . . . 6  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  -> 
( x  e.  ( K ... N )  <-> 
( K  -  1 )  <  x ) )
2815, 27orbi12d 690 . . . . 5  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  -> 
( ( x  e.  ( M ... ( K  -  1 ) )  \/  x  e.  ( K ... N
) )  <->  ( x  <_  ( K  -  1 )  \/  ( K  -  1 )  < 
x ) ) )
299, 28mpbird 223 . . . 4  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... N ) )  -> 
( x  e.  ( M ... ( K  -  1 ) )  \/  x  e.  ( K ... N ) ) )
30 elfzuz 10886 . . . . . . 7  |-  ( x  e.  ( M ... ( K  -  1
) )  ->  x  e.  ( ZZ>= `  M )
)
3130adantl 452 . . . . . 6  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... ( K  -  1 ) ) )  ->  x  e.  ( ZZ>= `  M ) )
32 elfzuz3 10887 . . . . . . . 8  |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K )
)
3332adantr 451 . . . . . . 7  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... ( K  -  1 ) ) )  ->  N  e.  ( ZZ>= `  K ) )
34 elfzuz3 10887 . . . . . . . . . 10  |-  ( x  e.  ( M ... ( K  -  1
) )  ->  ( K  -  1 )  e.  ( ZZ>= `  x
) )
3534adantl 452 . . . . . . . . 9  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... ( K  -  1 ) ) )  -> 
( K  -  1 )  e.  ( ZZ>= `  x ) )
36 peano2uz 10364 . . . . . . . . 9  |-  ( ( K  -  1 )  e.  ( ZZ>= `  x
)  ->  ( ( K  -  1 )  +  1 )  e.  ( ZZ>= `  x )
)
3735, 36syl 15 . . . . . . . 8  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... ( K  -  1 ) ) )  -> 
( ( K  - 
1 )  +  1 )  e.  ( ZZ>= `  x ) )
384recnd 8951 . . . . . . . . . . 11  |-  ( K  e.  ( M ... N )  ->  K  e.  CC )
396recnd 8951 . . . . . . . . . . 11  |-  ( K  e.  ( M ... N )  ->  1  e.  CC )
4038, 39npcand 9251 . . . . . . . . . 10  |-  ( K  e.  ( M ... N )  ->  (
( K  -  1 )  +  1 )  =  K )
4140eleq1d 2424 . . . . . . . . 9  |-  ( K  e.  ( M ... N )  ->  (
( ( K  - 
1 )  +  1 )  e.  ( ZZ>= `  x )  <->  K  e.  ( ZZ>= `  x )
) )
4241adantr 451 . . . . . . . 8  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... ( K  -  1 ) ) )  -> 
( ( ( K  -  1 )  +  1 )  e.  (
ZZ>= `  x )  <->  K  e.  ( ZZ>= `  x )
) )
4337, 42mpbid 201 . . . . . . 7  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... ( K  -  1 ) ) )  ->  K  e.  ( ZZ>= `  x ) )
44 uztrn 10336 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  x )
)  ->  N  e.  ( ZZ>= `  x )
)
4533, 43, 44syl2anc 642 . . . . . 6  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... ( K  -  1 ) ) )  ->  N  e.  ( ZZ>= `  x ) )
46 elfzuzb 10884 . . . . . 6  |-  ( x  e.  ( M ... N )  <->  ( x  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>=
`  x ) ) )
4731, 45, 46sylanbrc 645 . . . . 5  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( M ... ( K  -  1 ) ) )  ->  x  e.  ( M ... N ) )
48 elfzuz 10886 . . . . . . 7  |-  ( x  e.  ( K ... N )  ->  x  e.  ( ZZ>= `  K )
)
49 elfzuz 10886 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M )
)
50 uztrn 10336 . . . . . . 7  |-  ( ( x  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  x  e.  ( ZZ>= `  M )
)
5148, 49, 50syl2anr 464 . . . . . 6  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( K ... N ) )  ->  x  e.  ( ZZ>= `  M ) )
52 elfzuz3 10887 . . . . . . 7  |-  ( x  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  x )
)
5352adantl 452 . . . . . 6  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( K ... N ) )  ->  N  e.  ( ZZ>= `  x ) )
5451, 53, 46sylanbrc 645 . . . . 5  |-  ( ( K  e.  ( M ... N )  /\  x  e.  ( K ... N ) )  ->  x  e.  ( M ... N ) )
5547, 54jaodan 760 . . . 4  |-  ( ( K  e.  ( M ... N )  /\  ( x  e.  ( M ... ( K  - 
1 ) )  \/  x  e.  ( K ... N ) ) )  ->  x  e.  ( M ... N ) )
5629, 55impbida 805 . . 3  |-  ( K  e.  ( M ... N )  ->  (
x  e.  ( M ... N )  <->  ( x  e.  ( M ... ( K  -  1 ) )  \/  x  e.  ( K ... N
) ) ) )
57 elun 3392 . . 3  |-  ( x  e.  ( ( M ... ( K  - 
1 ) )  u.  ( K ... N
) )  <->  ( x  e.  ( M ... ( K  -  1 ) )  \/  x  e.  ( K ... N
) ) )
5856, 57syl6bbr 254 . 2  |-  ( K  e.  ( M ... N )  ->  (
x  e.  ( M ... N )  <->  x  e.  ( ( M ... ( K  -  1
) )  u.  ( K ... N ) ) ) )
5958eqrdv 2356 1  |-  ( K  e.  ( M ... N )  ->  ( M ... N )  =  ( ( M ... ( K  -  1
) )  u.  ( K ... N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1642    e. wcel 1710    u. cun 3226   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   RRcr 8826   1c1 8828    + caddc 8830    < clt 8957    <_ cle 8958    - cmin 9127   ZZcz 10116   ZZ>=cuz 10322   ...cfz 10874
This theorem is referenced by:  ballotlemfrceq  24035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-n0 10058  df-z 10117  df-uz 10323  df-fz 10875
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